Rounding of first-order phase transitions and optimal cooperation in scale-free networks

Details Rounding of first-order phase transitions and optimal cooperation in scale-free networks Rounding of first-order phase transitions and optimal cooperation in scale-free networks

2007-04-12

arxiv.org/abs/0704.1538v1

Phys. Rev. E76, 041107 (2007)

Physics

Condensed Matter

Statistical Mechanics

8 pages, 6 figures

Scientific paper

10.1103/PhysRevE.76.041107

We consider the ferromagnetic large-$q$ state Potts model in complex evolving networks, which is equivalent to an optimal cooperation problem, in which the agents try to optimize the total sum of pair cooperation benefits and the supports of independent projects. The agents are found to be typically of two kinds: a fraction of $m$ (being the magnetization of the Potts model) belongs to a large cooperating cluster, whereas the others are isolated one man's projects. It is shown rigorously that the homogeneous model has a strongly first-order phase transition, which turns to second-order for random interactions (benefits), the properties of which are studied numerically on the Barab\'asi-Albert network. The distribution of finite-size transition points is characterized by a shift exponent, $1/\tilde{\nu}'=.26(1)$, and by a different width exponent, $1/\nu'=.18(1)$, whereas the magnetization at the transition point scales with the size of the network, $N$, as: $m\sim N^{-x}$, with $x=.66(1)$.

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